In order to study the relationship between vibration and stiffness, the meshing stiffness and frequency spectrum of different modified tooth surfaces in the table are shown in Figure 1: the average meshing stiffness of conjugate tooth surface of hypoid gear is the largest, while the theoretical tooth surface modification of hypoid gear is the largest, and the average meshing stiffness is the smallest. The Fourier transform of the fluctuating part of the stiffness curve shows that:

① The meshing frequency amplitude and the third harmonic amplitude of the conjugate tooth surface of hypoid gear are larger, and the third harmonic amplitude of the optimized ease off tooth surface 2 is slightly smaller than the fundamental frequency amplitude;

② The fundamental frequency amplitude of the theoretical tooth surface, the optimal ease off tooth surface, the optimized ease off tooth surface 1 and the optimized ease off tooth surface 3 of the hypoid gear is greater than the harmonic component amplitude and the harmonic component amplitude is very small.

Based on the vibration optimization strength of the table, it can be seen that:

Different modified tooth surfaces	Load transmission error amplitude /%	Meshing line vibration acceleration /%
Optimal ease off tooth surface	65	15
Theoretical tooth surface	105	126
Optimize ease off tooth surface 1	106	22
Optimize ease off tooth surface 2	49	96
Optimize ease off tooth surface 3	56	33

① Under the rated working condition, the theoretical tooth surface Alte of hypoid gear basically has no change (105%). Due to excessive modification, the average meshing stiffness decreases too much, and the stiffness curve is mainly meshing frequency, and its vibration increases (126%);

② The Alte of tooth surface 1 of the optimized ease off hypoid gear is basically unchanged (106%), the average meshing stiffness decreases less, and the meshing frequency is mainly in the stiffness curve, while the vibration decreases to 22%;

③ The Alte of tooth surface 2 of optimized ease off hypoid gear decreases to 49%, the average meshing stiffness decreases less, and the meshing frequency and its third harmonic frequency are mainly in the stiffness curve, while the vibration acceleration changes little (96%);

④ The Alte of the tooth surface of the optimal ease off hypoid gear decreases to 65%, the average meshing stiffness decreases less, and the stiffness curve is mainly meshing frequency, while the vibration acceleration decreases to 15%;

⑤ The Alte of tooth surface 3 of optimized ease off hypoid gear is reduced to 56%, the average meshing stiffness is reduced less, the stiffness curve is mainly meshing frequency, and the vibration acceleration is reduced to 33%.

To sum up, when the amount of modification is too large, the stiffness decreases more and the normal vibration increases; After modification, the average meshing stiffness decreases less, and the normal vibration decreases obviously when the amplitude of stiffness curve decreases and the amplitude of harmonic wave is very small, while the normal vibration increases when the harmonic component in the stiffness curve is large; When the average meshing stiffness is similar, the shape of the stiffness curve is more sensitive to the vibration than the amplitude.

The conjugate tooth surface of hypoid gear resonates at 3200 and 5600 r/min, and the dynamic meshing force increases; The theoretical tooth surface modification is too large. Although the meshing force decreases at 5600 r/min, there is still a large resonance after 8000 r/min, and the meshing force increases; The normal meshing force of the optimized ease off tooth surface and the optimized ease off tooth surface 1 is significantly reduced within the whole speed range, as shown in Fig. 2 (a). Similarly, the conjugate tooth surface resonates at 3200 and 5600 r/min, and the normal vibration acceleration increases significantly; The resonance speed of the theoretical tooth surface decreases slightly and the vibration decreases near 5600 r/min; The vibration of the optimal ease off tooth surface and the optimal ease off tooth surface 1 decreases significantly near the resonance speed, but increases near 4500r/min, as shown in Figure 2 (b). The conjugate tooth surface vibration increases with the increase of load; The theoretical tooth surface vibrates with the increase of load, especially when the load is small, the actual coincidence degree is small, so the vibration is large; With the increase of load, the vibration of the optimal ease off tooth surface and the optimal ease off tooth surface 1 increases slowly, but decreases as a whole compared with the conjugate tooth surface and the theoretical tooth surface of the hypoid gear; The multi load Alte curve can basically reflect the variation trend of vibration with load, as shown in Figure 2 (c).